A stitching method is used to measure a test object having a large diameter. With the stitching method, shape measurement is repeated for partial regions of the test object by using a reference that is smaller than the test object, and data items obtained by measuring the shapes of the partial regions are stitched together by performing computation.
The stitching method may generate two types of measurement errors. First is an error that is generated when the position of a test object is moved due to mechanical instability while the partial regions are being measured. In the present description, this error will be referred to as a setting error. When there is a setting error, the shape data items obtained by measuring the partial regions include measurement errors that are different from each other. Second is an error inherent in the measurement system (optical system). In the present description, this error will be referred to as a system error. When there is a system error, the shape data items obtained by measuring the partial regions include measurement errors that are the same.
A sequential stitching method and a simultaneous stitching method are used to remove these errors. With the sequential stitching method, a reference shape data item is determined, the reference shape data item and shape data item adjacent to the reference shape data item are stitched together, and the stitched shape data item and the second adjacent shape data item are stitched together. All data items can be stitched together by repeating this process. However, a problem arises in that measurement errors accumulate. With the simultaneous stitching method, the shape data items are stitched together so as to minimize the accumulated error (see Weng W. Chow and George N. Lawrence, “Method for subaperture testing interferogram reduction”, OPTICS LETTERS, U.S.A., September 1983, Vol. 8, No. 9, pp. 468-470 (hereinafter referred to as NPL 1); Masashi Otsubo, Katsuyuki Okada, Jumpei Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms”, OPTICAL ENGINEERING, Japan, SPIE press, February 1994, Vol. 33 No. 2, pp. 608-613 (hereinafter referred to as NPL 2); and U.S. Pat. No. 6,956,657 (hereinafter referred to as PTL 1)). In general, it is said that the shape of a test object can be measured more accurately by using the simultaneous stitching method.
NPL 1 discloses a method for simultaneously correcting the setting error and the system error by using an equation. NPL 2 discloses a stitching method that is used when the shape data items overlap each other. PTL 1 discloses a method for simultaneously correcting the setting error and the system error when the shape data items overlap each other.
The equation used in NPL 1 is obtained in the case in which the shape data items for the partial regions do not overlap each other, and the equation is unsolvable in the case in which the shape data items overlap each other. In NPL 2, an equation for correcting only the setting error without consideration of the system error is solved, so that the system error cannot be corrected. PTL 1 does not describe an equation for stitching, and an optimization loop is repeated so that the accumulated error is minimized. In general, it is very difficult to solve an optimization problem. Therefore, the technology of PTL 1 requires sophisticated and complicated data processing and requires a long calculation time.